Is God Really Infinite?

Classical theism, with its identification of God with infinity, has developed a reputation for emphasizing divine transcendence to the point of making God nearly unknowable. The problem with this judgment is that infinity—as in, God is infinitely unknowable—does not admit to degrees. An infinite God is not like an unimaginably large number that we could count to if only we had enough time. Nor is an infinite God like the largest possible number we know, or at least know well enough to use in any practical way. That would be, according to the Guinness Book of World Records, Graham’s number, which has to do with the theoretical dimensions of the geometric shape known as a hypercube. Paradoxically, Graham’s number is at least as mysterious as the idea of infinity, since it exists only as a function of an extremely complex mathematical proof, and infinity, though hotly debated, is a fairly fixed idea—even if it is really nothing more than the idea of that which is unimaginable.

So which is harder to grasp: Graham’s number or the infinity of God? Graham’s number is so great that if each digit occupied a Planck volume (a unit too small for empirical use), the entire universe would not be big enough to contain it. Theologians in the tradition of classical theism claim that God is also greater than the known universe, but can they propose a rational way of demonstrating that God is greater than Graham’s number? We can know Graham’s number by describing how it functions (although really we are left with trusting that some very specialized mathematicians can describe this number to each other), but we can never know what that number actually is. If God is infinitely greater than that, then it seems like God is more like a random number picked from a range similar to Graham’s number. That number would be impossible to describe as well as to notate. That number, we could say, would be infinite all the way down. It would also be infinitely irrational, since it would be impossible to use it in any useful way.

God, of course, is not a number, no matter how big—nor is he an idea, no matter how impractical. Nevertheless, if I am right, mathematicians have managed to come closer to understanding infinity than theologians have. We can imagine Graham’s number no more than we can imagine an infinite being, but at least we have instructions about how to conceive of its possibility.

The closest theologians have come to instructions about how to understand God’s infinity is Anselm’s statement that God is “that than which nothing greater can be thought.” That proposition, however, is deceiving. It is not like a formula with rules for the intellectual moves that will lead our thought to a specific goal, which can be picked out among a number of alternatives. Anselm’s statement has a purely negative function. However far your thought can go, it will not reach God. Such a God might exist—philosophers still debate whether Anselm’s statement can serve as the basis for a proof of God’s necessary existence—but we cannot know anything positive about him. Attributing infinity to God and negative theology go hand in hand.

Anselm’s statement, however, can appear to be about something, which is why Erich Przywara, in Analogia Entis, is clearer than Anselm in saying that God is “beyond anything that can be thought.” Even that statement, however, is a bit misleading if it is interpreted as making a comparative point. An infinite God, Przywara argues, is not beyond what we know in any measurable degree. If we knew to what degree our knowledge of God falls short of its object, we would know something about who God really is. Our ideas fall so far short of God that we cannot even know how wrong they are. It follows that the clearer we make our conceptions of God—the more specific their reference or finite their form—the further God is beyond them. God’s infinity is infinitely receding, according to Przywara’s perspective. Knowing God is not just analogous to knowing what infinity is, since we can have some idea of that. No, the infinite God must be infinitely unknowable.

If it sounds strange to say that God’s infinity actually expands the closer we get to it, perhaps mathematics can again provide some illumination. Georg Cantor used set theory to demonstrate that infinite sets are not all the same and, thus, that there are a variety of infinities. While this seems to fly in the face of classical theism’s belief that God is uniquely infinite, it could also help to illuminate how it is that we can come to know an infinite God. God’s infinity, we could say, is simply different from the infinity that we can know something about through mathematical proofs. But if God is infinitely different from whatever it is we might think infinity is, then the concept of the infinite is absolutely equivocal when used theologically, and there is no sense in using it about God at all.

The alternative to this line of speculation is to return to Aristotle’s distrust of infinity (apeiron), an idea he found to be unsettling and not a little unthinkable, like a body without boundaries. Aristotle denied that anything that actually exists can be infinite, although he accepted a potential infinity, in the sense of a series that continues without any logical ending. Following Aristotle, perhaps the best we can mean by calling God infinite is that our knowledge and enjoyment of his presence will never be exhausted. God is like a hypercube whose dimensions, if ever mapped for the purposes of notation, would have no apparent numerical end. If so, then it is not quite accurate to say that God is infinite, but it would make some sense to say that our potential knowledge of God most certainly is.

Stephen H. Webb is a columnist for First Things. He is the author most recently of Mormon Christianity.

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